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Single Idea 19123
[filed under theme 5. Theory of Logic / K. Features of Logics / 3. Soundness
]
Full Idea
Gödel showed PA cannot be proved consistent from with PA. But 'reflection principles' can be added, which are axioms partially expressing the soundness of PA, by asserting what is provable. A Global Reflection Principle asserts full soundness.
Gist of Idea
If soundness can't be proved internally, 'reflection principles' can be added to assert soundness
Source
report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Halbach,V/Leigh,G.E. - Axiomatic Theories of Truth (2013 ver) 1.2
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.3
A Reaction
The authors point out that this needs a truth predicate within the language, so disquotational truth won't do, and there is a motivation for an axiomatic theory of truth.
Related Idea
Idea 19124
A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness [Halbach/Leigh]
The
40 ideas
from Kurt Gödel
9942
|
Gödel proved the classical relative consistency of the axiom V = L
[Gödel, by Putnam]
|
10868
|
The Continuum Hypothesis is not inconsistent with the axioms of set theory
[Gödel, by Clegg]
|
13517
|
If set theory is consistent, we cannot refute or prove the Continuum Hypothesis
[Gödel, by Hart,WD]
|
18062
|
Set-theory paradoxes are no worse than sense deception in physics
[Gödel]
|
8679
|
We perceive the objects of set theory, just as we perceive with our senses
[Gödel]
|
10271
|
Basic mathematics is related to abstract elements of our empirical ideas
[Gödel]
|
17751
|
Gödel proved the completeness of first order predicate logic in 1930
[Gödel, by Walicki]
|
21752
|
Prior to Gödel we thought truth in mathematics consisted in provability
[Gödel, by Quine]
|
17835
|
Gödel show that the incompleteness of set theory was a necessity
[Gödel, by Hallett,M]
|
10071
|
Second Incompleteness: nice theories can't prove their own consistency
[Gödel, by Smith,P]
|
17886
|
The limitations of axiomatisation were revealed by the incompleteness theorems
[Gödel, by Koellner]
|
19123
|
If soundness can't be proved internally, 'reflection principles' can be added to assert soundness
[Gödel, by Halbach/Leigh]
|
10621
|
Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme
[Smith,P on Gödel]
|
17888
|
The undecidable sentence can be decided at a 'higher' level in the system
[Gödel]
|
10132
|
There can be no single consistent theory from which all mathematical truths can be derived
[Gödel, by George/Velleman]
|
3198
|
Gödel showed that arithmetic is either incomplete or inconsistent
[Gödel, by Rey]
|
10072
|
First Incompleteness: arithmetic must always be incomplete
[Gödel, by Smith,P]
|
9590
|
Arithmetical truth cannot be fully and formally derived from axioms and inference rules
[Gödel, by Nagel/Newman]
|
11069
|
Gödel's Second says that semantic consequence outruns provability
[Gödel, by Hanna]
|
10118
|
First Incompleteness: a decent consistent system is syntactically incomplete
[Gödel, by George/Velleman]
|
10122
|
Second Incompleteness: a decent consistent system can't prove its own consistency
[Gödel, by George/Velleman]
|
10611
|
There is a sentence which a theory can show is true iff it is unprovable
[Gödel, by Smith,P]
|
10867
|
'This system can't prove this statement' makes it unprovable either way
[Gödel, by Clegg]
|
8747
|
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities
[Gödel, by Shapiro]
|
3192
|
Basic logic can be done by syntax, with no semantics
[Gödel, by Rey]
|
10041
|
Impredicative Definitions refer to the totality to which the object itself belongs
[Gödel]
|
21716
|
In simple type theory the axiom of Separation is better than Reducibility
[Gödel, by Linsky,B]
|
10035
|
Mathematical Logic is a non-numerical branch of mathematics, and the supreme science
[Gödel]
|
10038
|
A logical system needs a syntactical survey of all possible expressions
[Gödel]
|
10039
|
Some arithmetical problems require assumptions which transcend arithmetic
[Gödel]
|
10042
|
Reference to a totality need not refer to a conjunction of all its elements
[Gödel]
|
10043
|
Mathematical objects are as essential as physical objects are for perception
[Gödel]
|
10046
|
The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers
[Gödel]
|
10045
|
Impredicative definitions are admitted into ordinary mathematics
[Gödel]
|
17885
|
Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable
[Gödel, by Koellner]
|
10614
|
The real reason for Incompleteness in arithmetic is inability to define truth in a language
[Gödel]
|
10620
|
Originally truth was viewed with total suspicion, and only demonstrability was accepted
[Gödel]
|
17883
|
Gödel's Theorems did not refute the claim that all good mathematical questions have answers
[Gödel, by Koellner]
|
9188
|
Gödel proved that first-order logic is complete, and second-order logic incomplete
[Gödel, by Dummett]
|
17892
|
For clear questions posed by reason, reason can also find clear answers
[Gödel]
|